Cappell-Shaneson polynomials

S. Cappell and J. Shaneson constructed a pair of inequivalent embeddings of $(n-1)$-spheres in homotopy $(n+1)$-spheres for every square matrix of order $n$ with special properties (a Cappell-Shaneson matrix). A Cappell-Shaneson polynomial is the characteristic polynomial of a Cappell-Shaneson matrix. In this paper, we interpret part of the definition of Cappell-Shaneson polynomial as algebraic conditions on polynomials in terms of signed reciprocal polynomial and reduction modulo primes, and give complete lists of all Cappell-Shaneson polynomials of degrees $4$ and $5$. We construct several infinite series of Cappell-Shaneson polynomials of degree $6$.